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#install.packages("gsheet")
library(gsheet)
library(bayesrules)
library(rstanarm)
library(bayesplot)
library(tidyverse)
library(broom.mixed)
library(tidybayes)
## Bayesian Analysis of Pensacola Beach Loggerhead Sea Turtle Nesting Data Utilizing 10:1 Pseudo-Absence to Presence Ratio
Authors: Audrey Moore, MS & Laura Sikes, MPH<br> Instructor: Dr. Samantha Seals <br> Date: April 23, 2025 <br>
## Bayesian vs. Frequentist Approach
**Bayesian**
- Begins with existing belief (Prior) about probability distr. of hypothesized outcome.
- Introduces new data with a Likelihood distribution for hypothesized outcome.
- Bayes Theorem joins Prior and Likelihood resulting in Posterior, an updated probability distr. for hypothesized outcome.
- Strength of Prior and sample size determine influence on Posterior.
**Frequentist**
- Analyzes observed data.
- Results in a calculated best estimate of a parameter.
## Linear Regression (Normal)
| Frequentist | Bayesian |
|--------------------------------------|----------------------------------|
| Parameters treated as fixed constants | Parameters treated as random variables |
| Does not incorporate prior information | Incorporates prior information or beliefs |
| Confidence intervals for uncertainty | Credible intervals for uncertainty |
| Hypothesis testing using *p*-values | Hypothesis testing by interpreting posterior probabilities |
| Results express frequencies of estimates from multiple iterations | Results express probabilities about parameter values |
## Logistic Regression
- Bayesian logistic regression, similar to its frequentist counterpart, is used to model a binary categorical response variable (Y) given a set of predictor values (X). The resulting model estimates the log-odds of Y, which we can rewrite in terms of odds and probability.
- The Bayesian framework takes into account prior beliefs about the regression parameters, $\beta_0$ and $\beta_1$. Since these parameters can be any number on the real line, we are able to use the Normal distribution for both priors.
- Bayesian logistic regression involves updating these priors with the likelihood of the observed data so that we can make inferences about the relationship between X and Y.
## Unadjusted Analyses
data10to1 <- gsheet2tbl("https://docs.google.com/spreadsheets/d/1ARgHYUwclO5weZf5lE1f79V8otZdc4CLSpdCSmwP8Ts/edit?gid=1512758558#gid=1512758558")
test_wo_drop <- data10to1 %>%
na.omit()
data10to1 <- gsheet2tbl("https://docs.google.com/spreadsheets/d/1ARgHYUwclO5weZf5lE1f79V8otZdc4CLSpdCSmwP8Ts/edit?gid=1512758558#gid=1512758558") %>%
select(-do_not_use_FS, -do_not_use_BS)
test_w_drop <- data10to1 %>%
na.omit()
# Simulate the prior distribution
turtle_model_prior1 <- stan_glm(nested ~ beach_slope,
data = test_w_drop, family = binomial,
prior_intercept = normal(0, 2.5),
prior = normal(0, 2.5),
chains = 4, iter = 5000*2, seed = 120189,
prior_PD = TRUE)
# Update to simulate the posterior distribution
turtle_model1 <- update(turtle_model_prior1, prior_PD = FALSE)
# Simulate the prior distribution
turtle_model_prior2 <- stan_glm(nested ~ dune_ht,
data = test_w_drop, family = binomial,
prior_intercept = normal(0, 2.5),
prior = normal(0, 2.5),
chains = 4, iter = 5000*2, seed = 120189,
prior_PD = TRUE)
# Update to simulate the posterior distribution
turtle_model2 <- update(turtle_model_prior2, prior_PD = FALSE)
# Simulate the prior distribution
turtle_model_prior3 <- stan_glm(nested ~ foreshore_slope,
data = test_w_drop, family = binomial,
prior_intercept = normal(0, 2.5),
prior = normal(0, 2.5),
chains = 4, iter = 5000*2, seed = 120189,
prior_PD = TRUE)
# Update to simulate the posterior distribution
turtle_model3 <- update(turtle_model_prior3, prior_PD = FALSE)
# Simulate the prior distribution
turtle_model_prior4 <- stan_glm(nested ~ nest_dist,
data = test_w_drop, family = binomial,
prior_intercept = normal(0, 2.5),
prior = normal(0, 2.5),
chains = 4, iter = 5000*2, seed = 120189,
prior_PD = TRUE)
# Update to simulate the posterior distribution
turtle_model4 <- update(turtle_model_prior4, prior_PD = FALSE)
# Simulate the prior distribution
turtle_model_prior5 <- stan_glm(nested ~ nest_elev,
data = test_w_drop, family = binomial,
prior_intercept = normal(0, 2.5),
prior = normal(0, 2.5),
chains = 4, iter = 5000*2, seed = 120189,
prior_PD = TRUE)
# Update to simulate the posterior distribution
turtle_model5 <- update(turtle_model_prior5, prior_PD = FALSE)
# Simulate the prior distribution
turtle_model_prior6 <- stan_glm(nested ~ beach_slope + dune_ht + foreshore_slope + nest_dist + nest_elev,
data = test_w_drop, family = binomial,
prior_intercept = normal(0, 2.5),
prior = normal(0, 2.5),
chains = 4, iter = 5000*2, seed = 120189,
prior_PD = TRUE)
# Update to simulate the posterior distribution
turtle_model6 <- update(turtle_model_prior6, prior_PD = FALSE)
posterior_interval(turtle_model1, prob = 0.80)
posterior_interval(turtle_model2, prob = 0.80)
posterior_interval(turtle_model3, prob = 0.80)
posterior_interval(turtle_model4, prob = 0.80)
posterior_interval(turtle_model5, prob = 0.80)
posterior_interval(turtle_model6, prob = 0.80)
tidy(turtle_model1)
tidy(turtle_model2)
tidy(turtle_model3)
tidy(turtle_model4)
tidy(turtle_model5)
tidy(turtle_model6)
| Predictor | Model | 80% CI<br>(Lower) | 80% CI<br>(Upper) |
|:------------|:----------------------------------------------------------------------|-------------------|-------------------|
| <small>B. Slope</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -2.246 + 0.011\text{BS}$</small> | <small>-0.160</small> | <small>-0.169</small> |
| <small>D. Height</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -1.364 - 0.234\text{DH}$</small> | <small>-0.536</small> | <small>0.021</small> |
| <small>F. Slope</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -2.952 + 0.112\text{FS}$</small> | <small>0.017</small> | <small>0.205</small> |
| <small>N. Dist.</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -0.018 + 0.016\text{ND}$</small> | <small>-0.040</small> | <small>0.002</small> |
| <small>N. Elev.</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -0.483 + 0.306\text{NE}$</small> | <small>-0.891</small> | <small>-0.102</small> |
## Adjusted Analyses
<small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -0.963 - 0.074\text{BS} - 0.232\text{DH} + 0.178\text{FS} + 0.012\text{ND} - 0.980\text{NE}$ </small>
| Predictor | Adjusted 80% CI (Lower) | Adjusted 80% CI (Upper) |
|-----------------|------------------------:|------------------------:|
| Beach Slope | -0.314 | 0.144 |
| Dune Height | -0.555 | 0.038 |
| Foreshore Slope | 0.073 | 0.287 |
| Nest Distance | -0.039 | 0.057 |
| Nest Elevation | -1.822 | -0.100 |
## Diagnostics
neff_ratio(turtle_model1) #rhat and neff ratio
rhat(turtle_model1)
neff_ratio(turtle_model2)
rhat(turtle_model2)
neff_ratio(turtle_model3)
rhat(turtle_model3)
neff_ratio(turtle_model4)
rhat(turtle_model4)
neff_ratio(turtle_model5)
rhat(turtle_model5)
neff_ratio(turtle_model6)
rhat(turtle_model6)
mcmc_trace(turtle_model1, size = 0.1)
mcmc_trace(turtle_model2, size = 0.1)
mcmc_trace(turtle_model3, size = 0.1)
mcmc_trace(turtle_model4, size = 0.1)
mcmc_trace(turtle_model5, size = 0.1)
mcmc_trace(turtle_model6, size = 0.1)
mcmc_dens_overlay(turtle_model1)
mcmc_dens_overlay(turtle_model2)
mcmc_dens_overlay(turtle_model3)
mcmc_dens_overlay(turtle_model4)
mcmc_dens_overlay(turtle_model5)
mcmc_dens_overlay(turtle_model6)
## First Alternative Model Considered
# Simulate the prior distribution
turtle_model_prior7 <- stan_glm(nested ~ beach_slope + dune_ht + foreshore_slope + nest_dist + nest_elev + nest_dist*nest_elev,
data = test_w_drop, family = binomial,
prior_intercept = normal(0, 2.5),
prior = normal(0, 2.5),
chains = 4, iter = 5000*2, seed = 120189,
prior_PD = TRUE)
# Update to simulate the posterior distribution
turtle_model7 <- update(turtle_model_prior7, prior_PD = FALSE)
tidy(turtle_model7)
posterior_interval(turtle_model7, prob = 0.80)
<small>$\text{logit}(\pi_i) = -1.371 - 0.053\text{BS} - 0.208\text{DH} + 0.173\text{FS} + 0.042\text{ND} - 0.820\text{NE} -0.015\text{ND}\times\text{NE}$</small>
| Predictor | Adjusted 80% CI (Lower) | Adjusted 80% CI (Upper) |
|-------------------------------|-------------------:|-------------------:|
| Beach Slope | -0.298 | 0.184 |
| Dune Height | -0.528 | 0.060 |
| Foreshore Slope | 0.067 | 0.281 |
| Nest Distance | -0.036 | 0.128 |
| Nest Elevation | -1.741 | 0.109 |
| Nest Distance $\times$ Nest Elevation | -0.048 | 0.013 |
## Second Alternative Model Considered
# Simulate the prior distribution
turtle_model_prior8 <- stan_glm(nested ~ foreshore_slope + nest_elev + foreshore_slope:nest_elev,
data = test_w_drop, family = binomial,
prior_intercept = normal(0, 2.5),
prior = normal(0, 2.5),
chains = 4, iter = 5000*2, seed = 120189,
prior_PD = TRUE)
# Update to simulate the posterior distribution
turtle_model8 <- update(turtle_model_prior8, prior_PD = FALSE)
posterior_interval(turtle_model8, prob = 0.80)
| Predictor | Model | 80% CI<br>(Lower) | 80% CI<br>(Upper) |
|:------------|:----------------------------------------------------------------------|------------------:|------------------:|
| <small>B. Slope</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -2.246 + 0.011\text{BS}$</small> | <small>-0.160</small> | <small>-0.169</small> |
| <small>D. Height</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -1.364 - 0.234\text{DH}$</small> | <small>-0.536</small> | <small>0.021</small> |
| <small>F. Slope</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -2.952 + 0.112\text{FS}$</small> | <small>0.017</small> | <small>0.205</small> |
| <small>N. Dist.</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -0.018 + 0.016\text{ND}$</small> | <small>-0.040</small> | <small>0.002</small> |
| <small>N. Elev.</small> | <small>$\text{log}\left( \frac{\pi}{1-\pi}\right) = -0.483 + 0.306\text{NE}$</small> | <small>-0.891</small> | <small>-0.102</small> |
## Conclusions
Unadjusted Models:
- The 80% CI for foreshore slope is above 0, suggesting the chance of nesting increases as the foreshore slope increases.
- The 80% CI for nest elevation is below 0, suggesting the chance of nesting decreases as the nest elevation increases.
Adjusted Model:
- After adjusting for all predictors, we see similar results:
- Higher foreshore slope is associated with a higher probability of nesting
- Higher elevation is associated with a lower probability of nesting
## Code
</style>Bayesian Analysis of Pensacola Beach Loggerhead Sea Turtle Nesting Data Utilizing 10:1 Pseudo-Absence to Presence Ratio
Authors: Audrey Moore, MS & Laura Sikes, MPH
Instructor: Dr. Samantha Seals
Date: April 23, 2025
Bayesian vs. Frequentist Approach
Bayesian
- Begins with existing belief (Prior) about probability distr. of hypothesized outcome.
- Introduces new data with a Likelihood distribution for hypothesized outcome.
- Bayes Theorem joins Prior and Likelihood resulting in Posterior, an updated probability distr. for hypothesized outcome.
- Strength of Prior and sample size determine influence on Posterior.
Frequentist
- Analyzes observed data.
- Results in a calculated best estimate of a parameter.
Linear Regression (Normal)
| Frequentist | Bayesian |
|---|---|
| Parameters treated as fixed constants | Parameters treated as random variables |
| Does not incorporate prior information | Incorporates prior information or beliefs |
| Confidence intervals for uncertainty | Credible intervals for uncertainty |
| Hypothesis testing using p-values | Hypothesis testing by interpreting posterior probabilities |
| Results express frequencies of estimates from multiple iterations | Results express probabilities about parameter values |
Logistic Regression
Bayesian logistic regression, similar to its frequentist counterpart, is used to model a binary categorical response variable (Y) given a set of predictor values (X). The resulting model estimates the log-odds of Y, which we can rewrite in terms of odds and probability.
The Bayesian framework takes into account prior beliefs about the regression parameters, \beta_0 and \beta_1. Since these parameters can be any number on the real line, we are able to use the Normal distribution for both priors.
Bayesian logistic regression involves updating these priors with the likelihood of the observed data so that we can make inferences about the relationship between X and Y.
Unadjusted Analyses
| Predictor | Model | Unadjusted 80% CI (Lower) | Unadjusted 80% CI (Upper) |
|---|---|---|---|
| Beach Slope | \text{log}\left( \frac{\pi}{1-\pi}\right) = -2.246 + 0.011BS | -0.161 | 0.170 |
| Dune Height | \text{log}\left( \frac{\pi}{1-\pi}\right) = -1.364 - 0.234DH | -0.536 | 0.021 |
| Foreshore Slope | \text{log}\left( \frac{\pi}{1-\pi}\right) = -2.952 + 0.112FS | 0.017 | 0.205 |
| Nest Distance | \text{log}\left( \frac{\pi}{1-\pi}\right) = -0.018 + 0.016ND | -0.040 | 0.002 |
| Nest Elevation | \text{log}\left( \frac{\pi}{1-\pi}\right) = -0.483 + 0.306NE | -0.891 | -0.10 |
Adjusted Analyses
\text{log}\left( \frac{\pi}{1-\pi}\right) = -0.963 - 0.074\text{BS} - 0.232\text{DH} + 0.178\text{FS} + 0.012\text{ND} - 0.980\text{NE}
| Predictor | Adjusted 80% CI (Lower) | Adjusted 80% CI (Upper) |
|---|---|---|
| Beach Slope | -0.314 | 0.144 |
| Dune Height | -0.555 | 0.038 |
| Foreshore Slope | 0.073 | 0.287 |
| Nest Distance | -0.039 | 0.057 |
| Nest Elevation | -1.822 | -0.100 |
Diagnostics
First Alternative Model Considered
\text{log}\left( \frac{\pi}{1-\pi}\right) = -1.371 - 0.053BS - 0.208DH + 0.173FS + 0.042ND - 0.820NE - 0.015ND:NE
| Predictor | Adjusted 80% CI (Lower) | Adjusted 80% CI (Upper) |
|---|---|---|
| Beach Slope | -0.298 | 0.184 |
| Dune Height | -0.528 | 0.060 |
| Foreshore Slope | 0.067 | 0.281 |
| Nest Distance | -0.036 | 0.128 |
| Nest Elevation | -1.741 | 0.109 |
| Nest Distance \times Nest Elevation | -0.048 | 0.013 |
Second Alternative Model Considered
\text{log}(\frac{\pi}{1-\pi}) = -3.326 + 0.354FS + 0.075NE - 0.126FS:NE
| Predictors | 80% CI (Lower) | 80% CI (Upper) |
|---|---|---|
Foreshore Slope Nest Elevation Foreshore Slope x Nest Elevation |
0.129 -0.854 -0.268 |
0.607 1.045 0.005 |
Conclusions
Unadjusted Models:
- The 80% CI for foreshore slope is above 0, suggesting the chance of nesting increases as the foreshore slope increases.
- The 80% CI for nest elevation is below 0, suggesting the chance of nesting decreases as the nest elevation increases.
Adjusted Model:
- After adjusting for all predictors, we see similar results:
- Higher foreshore slope is associated with a higher probability of nesting.
- Higher elevation is associated with a lower probability of nesting.
APPENDIX A
CODE